Planned Comparisons in Factorial ANOVA

Planned Comparisons Following Factorial ANOVA

Following up a main effect in a factorial ANOVA is similar to following up a one-way ANOVA.
If the independent variable has only two levels, a planned comparison is unnecessary.
- With only two levels, significance indicates one level differs from the other. Examining the means reveals the direction of the effect.
Main effect comparisons can be pairwise or complex.

Compute S_c (the difference between two means).
Calculate the sums of squares for the comparison: SS{comparison} = \frac{Sc^2 * n}{2}, where n is the number of people sampled that comprise the means being compared.
The sums of squares for a pairwise comparison equals its mean squares because it's a single degree of freedom comparison.
Calculate the F ratio: F = \frac{MS{comparison}}{MS{error}}. MS{error} is the error term from the omnibus test, sometimes labeled as Ms or M/S_{error}.
If the calculated F exceeds the critical F, the difference between the pair is statistically significant.

Define coefficients to compare clusters of means.
Compute S_c using coefficients and means included in the analysis.
Use S_c to compute the sums of squares for the comparison, incorporating the number of people associated with each mean.
Even with combinations of means, this remains a single degree of freedom comparison, so the sums of squares equals the mean squares.

Analyzing interactions is more complex than main effects; a clear plan is essential.
The goal is to fully explain the interaction in an interpretable manner.
Planned comparisons should be determined in advance, but adjustments may be needed based on the data. If the plan is altered, it's important to adjust for running multiple comparisons.

A simple effect is the effect of one variable at a specific level of another variable.
- Example: The effectiveness of a taste manipulation (bitter vs. sweet water) specifically for conservatives.
Approach: Conduct a one-way ANOVA within one level of the factorial design.
Compute sums of squares for the simple effect comparison using the formula for between-groups ANOVA sums of squares: SS{simple \, effect} = \sum ni * (Mi - M{grand})^2, where ni is the sample size for each condition, Mi is the condition mean, and M_{grand} is the grand mean.
Compute mean squares. Unlike pairwise comparisons, degrees of freedom can be greater than one, so SS may not equal MS.
- Example: Comparing three taste conditions (sweet, sour, bitter) within conservatives would have 3 - 1 = 2 degrees of freedom.
Calculate the F ratio: F = \frac{MS{A{comp \, at \, level \, J}}}{MS{error}}, where A is one of the variables and J is a specific level of that variable, and MS{error} is the omnibus error term.

Pairwise comparisons within a subset of conditions in the design.
Compute S_c (simple difference between means).
Calculate sums of squares for the comparison using S_c. Since it's a single degree of freedom comparison, MS = SS.
Calculate the F ratio: F = \frac{MS{comparison}}{MS{error}}.

Planned comparisons within a level of the research design where groups of means are compared.
Compute S_c using contrast weights to define which groups to compare.
Calculate the sums of squares, incorporating sample size and contrast weights.
This will be a single degree of freedom comparison.
The F ratio is calculated as: F = \frac{MS{comparison}}{MS{error}}.

Report a measure of effect size, typically eta squared (\eta^2) for between-subjects designs.
\eta^2 = \frac{SS{effect}}{SS{total}}, where SS{effect} is the sums of squares for the specific effect (pairwise, complex, or simple), and SS{total} is the corrected total sums of squares from the SPSS output (excluding the intercept term sums of squares).